Pre-Algebra Examples

$\frac{3 q^{2} + 9 q + 6}{9 q^{2} - 36 q - 45}$

Step 1

Cancel the common factor of $3 q^{2} + 9 q + 6$ and $9 q^{2} - 36 q - 45$ .

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Step 1.1

Factor $3$ out of $3 q^{2}$ .

$\frac{3 (q^{2}) + 9 q + 6}{9 q^{2} - 36 q - 45}$

Step 1.2

Factor $3$ out of $9 q$ .

$\frac{3 (q^{2}) + 3 (3 q) + 6}{9 q^{2} - 36 q - 45}$

Step 1.3

Factor $3$ out of $3 (q^{2}) + 3 (3 q)$ .

$\frac{3 (q^{2} + 3 q) + 6}{9 q^{2} - 36 q - 45}$

Step 1.4

Factor $3$ out of $6$ .

$\frac{3 (q^{2} + 3 q) + 3 \cdot 2}{9 q^{2} - 36 q - 45}$

Step 1.5

Factor $3$ out of $3 (q^{2} + 3 q) + 3 \cdot 2$ .

$\frac{3 (q^{2} + 3 q + 2)}{9 q^{2} - 36 q - 45}$

Step 1.6

Cancel the common factors.

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Step 1.6.1

Factor $3$ out of $9 q^{2}$ .

$\frac{3 (q^{2} + 3 q + 2)}{3 (3 q^{2}) - 36 q - 45}$

Step 1.6.2

Factor $3$ out of $- 36 q$ .

$\frac{3 (q^{2} + 3 q + 2)}{3 (3 q^{2}) + 3 (- 12 q) - 45}$

Step 1.6.3

Factor $3$ out of $3 (3 q^{2}) + 3 (- 12 q)$ .

$\frac{3 (q^{2} + 3 q + 2)}{3 (3 q^{2} - 12 q) - 45}$

Step 1.6.4

Factor $3$ out of $- 45$ .

$\frac{3 (q^{2} + 3 q + 2)}{3 (3 q^{2} - 12 q) + 3 (- 15)}$

Step 1.6.5

Factor $3$ out of $3 (3 q^{2} - 12 q) + 3 (- 15)$ .

$\frac{3 (q^{2} + 3 q + 2)}{3 (3 q^{2} - 12 q - 15)}$

Step 1.6.6

Cancel the common factor.

$\frac{3 (q^{2} + 3 q + 2)}{3 (3 q^{2} - 12 q - 15)}$

Step 1.6.7

Rewrite the expression.

$\frac{q^{2} + 3 q + 2}{3 q^{2} - 12 q - 15}$

Step 2

Factor $q^{2} + 3 q + 2$ using the AC method.

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Step 2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $2$ and whose sum is $3$ .

$1, 2$

Step 2.2

Write the factored form using these integers.

$\frac{(q + 1) (q + 2)}{3 q^{2} - 12 q - 15}$

Step 3

Simplify the denominator.

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Step 3.1

Factor $3$ out of $3 q^{2} - 12 q - 15$ .

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Step 3.1.1

Factor $3$ out of $3 q^{2}$ .

$\frac{(q + 1) (q + 2)}{3 (q^{2}) - 12 q - 15}$

Step 3.1.2

Factor $3$ out of $- 12 q$ .

$\frac{(q + 1) (q + 2)}{3 (q^{2}) + 3 (- 4 q) - 15}$

Step 3.1.3

Factor $3$ out of $- 15$ .

$\frac{(q + 1) (q + 2)}{3 q^{2} + 3 (- 4 q) + 3 \cdot - 5}$

Step 3.1.4

Factor $3$ out of $3 q^{2} + 3 (- 4 q)$ .

$\frac{(q + 1) (q + 2)}{3 (q^{2} - 4 q) + 3 \cdot - 5}$

Step 3.1.5

Factor $3$ out of $3 (q^{2} - 4 q) + 3 \cdot - 5$ .

$\frac{(q + 1) (q + 2)}{3 (q^{2} - 4 q - 5)}$

Step 3.2

Factor $q^{2} - 4 q - 5$ using the AC method.

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Step 3.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 5$ and whose sum is $- 4$ .

$- 5, 1$

Step 3.2.2

Write the factored form using these integers.

$\frac{(q + 1) (q + 2)}{3 ((q - 5) (q + 1))}$

$\frac{(q + 1) (q + 2)}{3 (q - 5) (q + 1)}$

Step 4

Cancel the common factor of $q + 1$ .

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Step 4.1

Cancel the common factor.

$\frac{(q + 1) (q + 2)}{3 (q - 5) (q + 1)}$

Step 4.2

Rewrite the expression.

$\frac{q + 2}{3 (q - 5)}$